Cheiroballistra arms
= Introduction = While the generic form of arms is transmitted clearly from the original text and manuscript diagrams, there are a few important details that require discussion and real-life testing. Look here for closely related articles: * Description of the arms in the original manuscript * Pages in "Arms" category * Pages in "Cheiroballistra" category = Operating principle = The arms of the cheiroballistra are rather light, about 70 grams each. The cross-section of the steel bars in my rather high-power reconstruction is only 3.5x3.5mm. This works because as the cheiroballistra is shot, the elastic bowstring (made of nylon) cushions the initial impact after the arrow has just left. The residual energy in the arms makes them continue their rotation towards the curved (forward-facing) field frame bars. The bowstring stretches until the cones contact the curves in the bars. At this point the cone pivots against the curve and bends the torsion spring inwards, cushioning the rest of the blow. The only thing that will break the arms is a dry-fire, but this is generally the case anyways, regardless of arm design. = Materials = Everyone seems to agree that the bars in the arms were made from iron/steel. The hoops are also usually (e.g. Marsden 1971: 227) assumed to be of steel. Most also agree that the cones were of hardwood. Wilkins' (1995: 33) bad experience with breaking hardwood cones led him to start using bronze cones instead. That said, Wilkins' had increased the diameter of the springs (1995: 23-24) arbitrarily, because he assumed the cheiroballistra to be a winched weapon. He also lengthened the metal hooks far beyond the cones (1995: 33). Both of these increased the strain on the cones tremendously, so it's no wonder they broke in his tests, as noted earlier by Iriarte (2000: 59). If, for sake of the argument, we assume that a more durable material than wood was required for the cones, why make an elaborate composite construction in the first place? It would have been a lot simpler and cheaper to just forge the entire arm from steel and skip all the dovetail, hoop and tenon nonsense. In real-life tests the composite construction of wooden (elm) cones combined with fairly thin (<4mm) steel bars and soft steel hoops has worked extremely well with high energy shots. We can therefore fairly safely assume that those were the materials used in the original cheiroballistra. = Length of the cones = The text gives the ΙΑ (11) dactyls as the length of the cones, and based on practical tests there is no reason to doubt it. At one point, when hunting for "full draw" I theorized that the manuscript may have originally said ΙΔ (14) instead, which more or less allowed me to solve the draw length issue. However, I noticed later that I had made the little arch too long. If the little arch was of correct length, the 14 dactyl cones would not have enough clearance from the case. Moreover, when the spring cords are pretensioned enough using the stretcher, the washers don't have to be rotated much to generate enough power. This in turn allows drawing the shorter 11 dactyl arms far enough without causing the torsion spring to stack. = Length of the bars = First ambiguity in the text is the length of the (metal) bars of the arms, which is not given. The codex M version of the diagram (see Schneider 1906: 162; Wilkins 1995: 32) show the bars extending quite far beyond the end of the cones, in the codex V version not so much (see Wilkins 1995: 32). Due the above ambiguity scholars such as Marsden (1971: 226-227) and Wilkins (1995; 2003) were able to make the metal hooks very long without violating any instructions given in the text. Wilkins had to do this because he had issues with too short draw length, which in turn were caused by placing of the little ladder too close to the front of the case and not against the projecting block where it almost certainly belongs. From engineering point of view extending the metal bars significantly beyond the cones is very unwise for two reasons, each of which "feeds" the other and amplifies the problem. First, extending them places too much stress on the ends of the metal hooks, which would have otherwise been supported by the rigid wooden arm. In other words, they become a lot more prone to bending and therefore they have to be made much thicker and thus heavier, which indirectly weakens the ends of the wooden cones whose dimensions are known from the manuscript. Also, as the arms are only a method by which energy is transmitted to the bowstring and to the bolt, they should be as light as possible. The heavier the ballista arms are, the more energy they waste, especially if their forward momentum is stopped by the pads in the curved field-frame bars and not the bowstring, which would otherwise transfer some of the wasted energy into the bolt. Second, as Iriarte (2000: 59) points out, extending the bars far beoynd the cone adds shearing stress to it's end, which is already it's weakest point. This probably requires further explanation. Imagine the bowstring trying to pull the bar at 90 degree angle away from the cone: the hoop gets pulled upwards, and the end of cone gets pushed downwards. So, all force applied to the bar is concentrated to these two points and the middle part of the bar is free to rise upwards. Extending the bars far beyond the cones increases the pressure experienced by the two support points, very likely resulting in cone breakage at it's narrow end if too much force is applied. Due to the above issues the only reasonable option is to make the bar just long enough to allow the base of the hook to pass the tip of the cone, and to be supported by it. The actual length of the bar depends on how the rear-end is attached (option #1 and #2), so no absolute measurement can be given. If one really needs longer arms then it is best to make the cones and the bars longer, so that the overall thickness of the bar needs not be increased. = Shape of the channel = Shape of the channel in the cones is also a matter of debate. On the other hand, in all versions of manuscript diagrams (M and V) the channel has a distinct dovetail shape. However, as mentioned on the translation of cheiroballistra page and also by Iriarte (2000: 59) P.H. talks about a square or a rectangular channel, not a dovetailed one. So, based on the cheiroballistra manuscript alone both rectangular and dovetail-shaped channels would be acceptable. There are technical reasons why a rectangular channel makes more sense than a dovetailed one. First, cutting a rectangular channel and forging a rectangular bar is trivial, and an extremely tight fit is very easy to accomplish. In practical tests rectangular bar in rectangular channel has worked perfectly with absolutely no issues. Bindings can be used to prevent the bar from rising from the channel during pullback; That said, practical tests have shown that this is not an issue with straight arms. Even if the cones are reflexed (curved outwards) for whatever reason, the bar can be secured to the bottom of the groove with a small amount of linen bindings. A good dovetailed channel could be made with some effort by attaching two strips of wood with correct angles to the cone proper. However, the angles of the strips have to match those of the bar very closely. Alternatively the channel could be shaped using a very small and fine saw and a tiny chisel. Forging a dovetailed bar is possible, but fairly difficult without a special form. Filing the bar to the correct dovetail shape works, but takes a lot of time. So, in a nutshell, a tightly fitting dovetailed construction requires lots of time and care, and does not really yield any benefits compared to a rectangular construction; instead, it actually makes the entire construction brittle: during pullback the middle part of the bar rises upwards. This means that if a dovetailed bar is tightly fitted into the channel, the bar will rip apart the sides of the channel and possibly result in arm breakage. This issue can be mitigated by bindings, but if bindings have to used, there is no reason to not make a much simpler rectangular channel. There is also the option that the bars were rectangular in cross-section, but became progressively narrower towards the hooks. Forging a tapering, rectangular bar is fairly easy. The tapering channel and bars allow making light and fast arms, which will also be strong. However, a tapering bar and a channel will require some extra work compared to a simple rectangular bar and channel and may not be worth the effort in the end. It seems that the only reasonable option is to make the channel and the bar rectangular and of uniform dimensions along their entire length. If extreme velocities are required, then a tapering bar and channel can be used, but this requires a lot more care for probably little gain. = Form of the hooks in the bars = Practical tests have shown that the hooks in the bars will vertically auto-center and correct their orientation during pullback if the hook extends beyond the belly of the arm. This allows the use of cones of circular or heavily rounded, instead of flat, cross-section. = Attaching the bars to cones = The manuscript text says that the channel for the bar runs the entire length of the cone. However, it is also clear from manuscript text and diagrams that the bar did not fill the channel in the cone entirely; instead, it stopped at the middle, apparently near the torsion spring. At this point the cone and the bar were joined using a hoop of some sort. Another attachment point (a socket?) was right at the tip of the cone. As mentioned earlier, the cones taper towards the tips - this has important implications as we'll see later on. Optimally we'd need something that works in practice yet does not require changes to the given instructions. In fact, we have several problems to solve and we'll discuss each of these in turn. What material was used for the hoop near the base of the cone? As discussed elsewhere, the arm does not need very much protection against the impact with the field-frame bars. This means that we don't have to use metal hoops or plating to protect the cones. Therefore a fairly thick open hoop made from soft ''steel is the simplest possible solution for keeping the arm rigidly in the cord bundle. Note that the steel in the hoop can't be springy at all, because it needs to be squeezed ''in the wood fibers in the cone, so that it does not move when the arm is rotated and the spring cords start forcing it to the side. How do we keep the bar from sliding towards the tip of the cone? This problem can be solved by extending the bar over the base of the cone and bending it against it; this approach has the advantage that should the cone break, the arm as a whole stays in one piece instead of exploding into bits and pieces. Other equally valid methods include bending the bar forward around the hoop and extending the bar slightly beyond the hoop, drilling a small hole through it and fixing it with a small nail. Yet another (untested) way is to file a small notch for the hoop into the bar: when the hoop is tightened the bar is locked rigidly in place. Bending the end of the bar downward into the cone seemed to weaken the arm considerably, even though in theory it should have worked as the hole was near the neutral (non-bending) zone. How do we prevent the non-hooked end of the bar from rising up from the channel? This problem is solved easily by extending the bar inside the spring bundle. Any type of hoop or bindings will also hold the bar in place. How do we make the tip of the cone rigid? Looking at the cheiroballistra manuscript is seems clear that the socket (τόρμος) was used to bind the tip of the cone and the bar together. The exact form of the socket is not described, though, which has resulted in a number of differing constructions. I personally think that Iriarte (2000: 59) is by far closest when suggesting that the cone and the bar were simply wrapped together with a cord. However, he made (2000: 60) the ends of the cones protude from the cones proper as tenons, thus weakening the cone's tip. As discussed in the Translation of Cheiroballistra article, the term τόρμος that describes the socket means "any hole or socket, in which a pin or peg is stuck". So, in my opinion, there is no need to add a tenon; instead, we can add a socket formed by a second hoop at the end of the cone. In other words the end of the cone becomes the socket, into which the bar is pushed. This interpretation finds some support from the manuscript diagrams, too. The best material is probably sinew thread, although linen thread glued to place has worked extremely well in practice. The socket could have been formed by small metal ring also. How do we keep the cone from slipping through the torsion spring thin end first? This problem is solved easily by making the entire cone taper towards the tip, as the base is thicker it can't squeeze itself through a torsion spring that's pretensioned significantly. This is what's implied in the text and has shown to work very well in practice. Also, when the torsion spring is pretensioned to realistic power levels, the cords actually sink into the cone, thus helping to prevent any cone movement. How do we keep the cone from slipping through the torsion spring thick end first? As the cone becomes progressively thinner towards the tip, the pressure of the spring cords tends to slowly squeeze the arm out of the cord bundle the thick end first. This has to be prevented somehow, and a fairly thick. open hoop of round or rectangular cross-section made from fits this purpose perfectly. The hoop works best if made from soft steel, so that it can be tightened with a pair of heavy pliers or using a vise, so that the hoop actually forces itself into the wood fibers. It is also beneficial - or even necessary - to file small, round grooves to the sides of the cone for the hoop. Taking away a small amount of wood from the sides does not weaken the cone much, but still ensure that the hoop does not budge, when the spring cords try to displace it during pullback. Soft steel hoops have proven themslves in practice. Note that if the hoop is too thin, the spring cords will climb over it - about 6 mm thick round hoop seems to prevent this adequately. Increasing linear pretension also reduces the risk of cords slipping over the hoops. Although simple bindings made from strong thread (e.g. sinew or linen) would be significantly lighter, weight does matter much so close to the spring bundle: for further discussion on this subject look at the Ballista performance article. The primary problem with a hoop made from wrapped thread or sinew is that it doesn't have the bulk required to keep the spring cords in check, unless multiple layers are glued together into a thick bundle. How to prevent the arm being damaged when it hits the curved field-frame bar? The impact of the arms and the field-frame bars is not especially violent even in the worst case, that is when the ballista is dry-fired. For example, a dry-fire at fairly high power level ripped the hooks and the bowstring apart, but did no visible damage to the cones. This makes sense, because the rotational velocity of an arm is fairly small where it contacts the field-frame bar. So, when the slowly moving arm contacts the curve in the field-frame bar, the curve becomes a pivot point for the arm, which then uses up most of it's energy trying to bend the torsion spring to the side. Rounding the corners of the curve is still a good idea. The original revelation has led me to move away from using long, protective metal hoops towards lightly cushioning the the field-frame bar instead. One option would be to make the bar in the arm rise up from it's groove so that it would contact the field-frame bar first. = Tillering the arms = The topic of tillering and bow design is discussed in depth in the the Crossbow building wiki. Although the cones in cheiroballistra are not limbs of a bow, the bow design principles can and should be applied to them to produce optimal results. Cones where stress is concentrated on certain parts (e.g. the middle) will break at much lower loads than arms which are tillered so that the load is spread evenly along the entire length of the cone. In other words properly tillered arms allow can safely bear the same load as heavier arms that are incorrectly tillered. The cheiroballistra in particular requires arms that are tillered as shown below unless one decides to increase the dimensions for the cones arbitrarily. Simple planed cones that taper linearly in both width and in thickness will bend too much in the middle, not only in theory but also in practice. With high power levels linearly tapered cones will break. The roughly correct tiller looks like this (all dimensions in millimeters, click for full-size image): The dimensions are based on the following premises: # The leverage reduces from 1.0 to 0.0 times maximum as we move towards the tips of the arms. This is not 100% correct, but close enough for practical purposes. # Reducing width by n percent increases bend by n percent (at any given point) # Halving thickness increases bend to eight-fold (2^3, i.e. to cube root) This results in a width taper that is linear, primarily because it is the simplest option. Thickness taper, however, is elliptical, i.e. the rate at which arm thickness is reduced increases when moving towards the tip. Note, however, that there are four places where the optimal tiller is not followed: * At the hoop the cones are thicker to compensate for the small grooves for the hoop at the sides, and for the crushing effect of the hoop itself. * Near the outer torsion spring half the cone is made thicker, because the cords actually crush the topmost wood fibers, weakening that area. * Near the tip of the cone the minimum thickness is left at 0.5 dactyls (~10mm). * The base of the cone is rounded, so that there is enough clearance of the field frame bar. When finished, the arms look like this: Notice the glued-on linen bindings that act as the socket, as well as the center linen binding which was deemed necessary because one of the arms is reflexed. Lots of details about practical experiments are available in Samuli's blog: * Getting closer * Plans for improved cones * Elliptic tiller on arms * Back in business Category:Backup Category:Cheiroballistra Category:Practical Category:Theoretical Category:Arms